there really isn't any "ongoing maintenance" needed usually.
to move from your current host to a new one, only basic web skills are needed.
you need to know how to use ftp and how to make database backups and restores.
this might help:
https://www.phpbb.com/support/docs/en/3 ... or-domain/
as for finding a new host. you can search for one here:
you don't need a huge server or a dedicated server etc. just a normal shared hosting account with the basics.
when you get the backups from forumatic you will have a copy of your files and a database backup file.
that is all you need to transfer to a new host and install your board.
future updates will be a simple 5 minute job , installing extensions and/or styles, if you need or want them, will be a 5 minute job etc.